A sample of low-redshift BL Lacertae objects. II. EVN and MERLIN data and multi-wavelength analysis M. Giroletti (1 and 2), G. Giovannini (1 and 2), G. B. Taylor (3 and 4), R. Falomo (5) ((1) INAF Institute of Radioastronomy, (2) Univ. of Bologna, (3) Univ. of New Mexico, (4) NRAO Socorro, (5) INAF Observatory of Padua) Accepted by ApJ. 17 pages. We present new radio observations of 9 members of a sample of 29 nearby (z < 0.2) BL Lac objects. The new data have been obtained with the European VLBI Network and/or the MERLIN at 1.6 and 5 GHz and complement previous observations. For one object, the TeV source Mrk 421, we also present deep multi-epoch VLBA and Global VLBI data, which reveal a resolved diffuse jet, with clear signatures of limb brightening. We use the new and old data to estimate physical parameters of the jets of the sample from which the subset with new radio data is drawn. We derive Doppler factors in the parsec scale radio jet in the range ~2 < delta < ~9. Using HST data, we separate the contribution of the host galaxy from that of the active core. From the measured and de-beamed observables, we find a weak correlation between radio power and black hole mass, and a tight correlation between radio and optical core luminosities. We interpret this result in terms of a common synchrotron origin, with little contribution from a radiatively efficient accretion disk. The BL Lacs in our sample have de-beamed properties similar to low power radio galaxies, including the fundamental plane of black hole activity.